LimitsDerivativesIntegralsAppendixMath Calculus ReviewCHSN Review ProjectContentsLimits 3Derivatives 14Integrals 33Appendix 42This review guide was written by Dara Adib. Portions of the “Limits” and “Derivatives” chapters arebased off the Calculus Wikibook available on the Internet at http://en.wikibooks.org/wiki/Calculus. CHSN Review Project contributors Dara Adib and Paul Sieradzki contributed to the“Limits” section of the Calculus Wikibook.This is a development version of the text that should be considered a work-in-progress.This review guide is developed by the CHSN Review Project. To download this review guide andother review guides, visit chsntech.org.Copyright © 2008-2009 Dara Adib and other contributors to the Calculus Wikibook. This is a freelylicensed work, as explained in the Definition of Free Cultural Works (freedomdefined.org). Ex-cept as noted under “Graphic Credits” on this page, it is licensed under the Creative CommonsAttribution-Share Alike 3.0 Unported License. To view a copy of this license, visithttp://creativecommons.org/licenses/by-sa/3.0/ or send a letter to Creative Commons,171 Second Street, Suite 300, San Francisco, California, 94105, USA.This review guide is provided “as is” without warranty of any kind, either expressed or implied.You should not assume that this review guide is error-free or that it will be suitable for the particularpurpose which you have in mind when using it. In no event shall the CHSN Review Project beliable for any special, incidental, indirect or consequential damages of any kind, or any damageswhatsoever, including, without limitation, those resulting from loss of use, data or profits, whether ornot advised of the possibility of damage, and on any theory of liability, arising out of or in connectionwith the use or performance of this review guide or other documents which are referenced by orlinked to in this review guide.Graphic Credits• Figure 0.1 on page 24 is a public domain graphic by Inductiveload:http://commons.wikimedia.org/wiki/File:Maxima_and_Minima.svg• Figure 0.2 on page 26 is a public domain graphic by Inductiveload:http://commons.wikimedia.org/wiki/File:X_cubed_(narrow).svg2LimitsThis chapter was originally designed for a test on limits administered by Jeanine Lennon to her Math12H (4H/Precalculus) class on April 2, 2008. It was later updated with an “Addendum” section (page12) for a test on limits administered by Jonathan Chernick to his AP1Calculus BC class on September18, 2008.IntroductionA limit looks at what happens to a function when the input approaches, but does not necessarilyreach, a certain value. The general notation for a limit is below.limx→cf(x) = LThis is read as “the limit of f(x) as x approaches c is L.”Informal Definition of a LimitL is the limit of f(x) as x approaches c. The value of f(x) comes close to L when x is close (but notnecessarily equal) to c. It can be represented by either of the following forms, with the former beingfar more common.• limx→cf(x) = L• f(x) →L as x →cRulesNow that a limit has been informally defined, some rules that are useful for manipulating a limit arelisted.IdentitiesThe following identities assume limx→cf(x) = L and limx→cg(x) = M. Using these identities, other rulescan be deduced.1AP is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, thisproduct.3Scalar MultiplicationA scalar is a constant. When a function is multiplied by a constant, scalar multiplication is performed.limx→ckf(x) = k · limx→cf(x) = kLAdditionlimx→c[f(x) + g(x)] = limx→cf(x) + limx→cg(x) = L + MSubtractionlimx→c[f(x) − g(x)] = limx→cf(x) − limx→cg(x) = L − MMultiplicationlimx→c[f(x) ·g(x)] = limx→cf(x) · limx→cg(x) = L ·MDivisionlimx→cf(x)g(x)=limx→cf(x)limx→cg(x)=LM, where M 6= 0Constant RuleThe constant rule states that if f(x) = k is constant for all x, then the limit as x approaches c must beequal to k.limx→ck = kIdentity RuleThe identity rule states that if f(x) = x, then the limit as x approaches c is equal to c.limx→cx = c4Power RuleThe rule for products many times results in determining the power rule.limx→cf(x)n=limx→cf(x)nFinding LimitsIf c is in the domain of the function and the function can be built out of rational, trigonometric,logarithmic and exponential functions, then the limit is simply the value of the function at c.If c is not in the domain of the function, then in many cases (as with rational functions) the domainof the function includes all of the points near c, but not c. An example would be if one wanted tofind limx→0xx, where the domain includes all real numbers except 0. In that case, one would want tofind a similar function, with the hole filled in. The limit of this function at c will be the same, whilethe function is the same at all points not equal to c. The limit definition depends on f(x) only at thepoints where x is close to c but not equal to it. And since the domain of the new function includesc, one can now (assuming it’s still built out of rational, trigonometric, logarithmic and exponentialfunctions) just evaluate the function at c as before.In the above example, this is easy; canceling the x’s gives 1, which equalsx/x at all points except 0.Thus, limx→0xx= limx→01 = 1. In general, when computing limits of rational functions, it’s a good idea tolook for common factors in the numerator and denominator.Does Not ExistNote that the limit might not exist at all. There are a number of ways in which this can occur.Not Same from Both SidesA left-handed limit is different from the right-handed limit of the same variable, value, and function.Since, the left-handed limit 6= right-handed limit, the limit does not exist. This includes cases in whichthe limit of a certain side does not exist (e.g. limx→2√x − 2, which has no left-handed limit).GapThere is a gap (more than a point wide) in the function where the function is not defined. As anexample, in f(x) =√x2− 16, f(x) does not have any limit when −4 ≤ x ≤ 4. There is no way to“approach” the middle of the graph. Note also that the function also has no limit at the endpoints ofthe two curves generated (at x = −4 and x = 4) since limits from both sides do not exist.5JumpIf the graph suddenly jumps to a different level, there is no limit. This is illustrated in the floorfunction (in which the output value is the greatest